Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,17)(11,12,15,19)(13,16,22,24)(18,20,21,23)(25,26,32,36)(27,37,39,30)(28,38,40,33)(29,35,31,34);(3,7)(6,8)(9,11)(10,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,27,32,39)(26,30,36,37)(28,35,40,34)(29,33,31,38);(1,2)(3,6)(4,5)(7,8)(11,15)(12,19)(18,21)(20,23)(25,28,32,40)(26,33,36,38)(27,34,39,35)(29,37,31,30);(9,12,14,19)(10,15,17,11)(13,20,22,23)(16,21,24,18)(25,29,32,31)(26,34,36,35)(27,33,39,38)(28,37,40,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,23)(22,24)(25,30)(26,27)(28,29)(31,40)(32,37)(33,34)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,24)(19,23)(25,31,32,29)(26,35,36,34)(27,38,39,33)(28,30,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746483);